Is there an equivalent to Remove Doubles for Bezier curves?
There is no equivalent to Remove Doubles for Bezier Curves. This is likely due to the difference in the nature of Curves versus Meshes. As Denis stated, if the two points have different handle positions, how should this be, um, handled? ;-) Should the handle positions be averaged together? Also, Remove Doubles can just merge vertices based on a distance threshold no matter how many edges are connected to it, but Curves are fundamentally different since a point cannot have more than two segments connected to it. Simply put, Curves are linear while Meshes are non-linear. 

This doesn't necessarily mean such a feature couldn't be implemented, but it would have limited use cases, and for non-identical doubles it would have to make reaching assumptions about how you want your points merged. For example, if your Curve doesn't cross itself you may be alright, but if it does it could easily end up getting merged at the intersection and the path rerouted in an unexpected way. Here's a screen shot to illustrate what could happen hypothetically:



In conclusion, it could be useful in cases where the doubles lie right on top of one another, but at higher distance thresholds could produce unexpected results. Such a feature could be useful for certain situations, such as for removing points lying on top of one another that you can't see. 

For now, you could find such disconnected points by selecting at least one point on each of your curve segments you intend to keep, using Select Linked CtrlL, Hiding them H, then Select All A and Delete X the disconnected left-over points. Then Un-Hide AltH the segments you will keep.